Hilb becomes a *-Category


Next: Hilb becomes a Monoidal Up: Categories in Quantum Mechanics Previous: Category Hilb (Hilbert spaces)

Hilb becomes a *-Category

Generally speaking in order to make the category Hilb into a *-Category one should define for each operator $ T:H\rightarrow H^{'}$ its adjoint by $ T^*:H^{'}\rightarrow H$ such that the inner product can now be defined as:

$\displaystyle \langle T^*\Psi,\Phi\rangle=\langle\Psi,T\Phi\rangle$    

From the definition of the *-Category (2.3) it follows that the identification of the adjoint of an operator with $ T^*$ insures that the properties of operators in Quantum Mechanics are respected, i.e.
  • $ Id^*_H=Id_H$
  • $ (T_1T)^*=T^*T_1^*$
  • $ T^{**}=T$
  • $ (T+T_1)^* = T* + T_1*$
  • $ (aT)^* = a* T^*$ where $ a\in{\mathchoice { \setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$
The above definition of adjoint enables us to define a one to one correspondence between each vector in a Hilbert space and a corresponding operator such, that the, inner product turns out to be a function from the vector space of complex numbers to itself, as required form Quantum Mechanics. To better understand this let us define a vector $ \Psi\in H$ as follows $ T_{\Psi}:{\mathchoice { \setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}\rightarrow H$ with $ T_{\Psi}(1)=\Psi$,1.3 then it is easy to deduce that for each $ \Psi\in H$ there exists a unique $ T_{\Psi}$ and vice versa. It then follows that the inner product becomes

$\displaystyle \langle\Psi,\Phi\rangle=T^*_{\Psi}T_{\Phi}= {\mathchoice { \setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}\stackrel{T_{\Phi}}{\rightarrow}H\stackrel{T^*_{\Psi}}{\rightarrow}{\mathchoice { \setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$    

as required
Cecilia Flori 2006-12-04